State University has increased its tuition for in-state and out-of-state students in each of the past five years to offset cuts in its budget allocation from the state legislature. The university administration always thought that the number of applications received was independent of tuition; however, drops in applications and enrollments the past two years have proved this theory to be wrong. University admissions officials have developed the following relationships between the number of applicants who accept admission and enter State University (x) and the cost of the tuition per semester (t):
x1 = 21000 - 12t_1(in-state)
x2 = 35000 - 6t_2 (out-of-state)
The university would like to develop a planning model that will indicate the in-state and out-of-state tuitions, and the number of students that could be expected to enroll in the freshman class. The university doesn’t have enough classroom space for more than 1,400 freshmen, and it needs at least 700 freshmen to meet all the classes. The university knows from historical data that approximately 55% of all in-state freshmen will want to live in the school dormitories, and 72% of all out-of-state students will want to live in the dormitories, and there will be at most 800 dormitory rooms available for freshmen (each room can accommodate two people).
The university also wants to make sure that it maintains high academic standards in its admissions decisions. It knows from historical data that the average SAT score for an in-state student is 960 and the average SAT score for an out-of-state student is 1,150. The university wants the entering freshmen class to have an average SAT score of at least 1,000.
State University is a state-supported institution, so the state legislature wants to make sure that the university doesn’t enroll just out-of-state students, because they pay more tuition and they have better SAT scores. Thus, the government has instituted a policy that no more than 55% of the entering freshmen class can be out-of-state students.
Develop and solve a nonlinear programming model for State University that will indicate the tuition the university should charge, the total tuition, and the number of in-state and out-of-state students it can expect with these tuition rates.